Simulated Interactions Between Symmetric Tropical Cyclone-like ...
Simulated Interactions Between Symmetric Tropical Cyclone-like Vortices Peter M. Finocchio Rosenstiel School of Marine and Atmospheric Science University of Miami, Miami, FL April 30, 2013
1. Introduction
tween two symmetric tropical-cyclone like vortices without the constraints of quasi-geostrophy. In order to simulate vortex interactions in the real atmosphere, we enlist the fully-compressible Weather Research and Forecast (WRF) model. A variety of experimental vortices of variable depth are simulated in order to understand the role of vortex depth in the merger process. Following von Hardenburg et al. (2000), we can anticipate how the critical merger distance might behave in 2D vs. 3D stratified flow using the Green’s Function. In 2D, the induced flow by a point vortex decays radially as log(r). However, in 3D stratified flow, the induced flow by a point of potential vorticity (PV) decays as a modified Bessel function of the second kind, implying more localized interactions. 3D vortex interactions are also sensitive to the degree of stratification: Stronger stratification increases the deformation radius and thus the extent of the induced flow field by a vortex. As taller vortices better approximate a 2D vortex (which has infinite depth), we expect the critical merger distance to increase with vortex depth. Both von Hardenburg et al. (2000) and Dritschel (2002) encounter such behavior for taller vortices, but some confusion arises on whether the 2D barotropic limit is indeed a limit on critical merger distance. It is precisely this sort of discrepancy that we wish to clarify in this study.
Vortex merger and interaction has been studied extensively from a theoretical fluid dynamics perspective. How such interactions unfold in the real atmosphere, however, remains unclear. Nonetheless, twodimensional vortex dynamics is a natural starting point from which to understand atmospheric vortex interactions. Melander et al. (1987) first illustrated the so-called axisymmetrization of an elliptical twodimensional patch vortex as a process in which filamentation of the vortex core disrupts the symmetry of the flow field. As a result, stream lines can cross vorticity contours so as to decrease the aspect ratio of the elliptical vortex, ultimately forcing it toward near axisymmetry. Melander et al. (1988) considered vortex merger an axisymmetrization process, and established a critical merger distance for two patch vortices of the same size and strength (i.e. “symmetric” vortices) ≈ 3.2 times the initial vortex radius (R). We will henceforth refer to 3.2R as the 2D limit for vortex merger. Subsequent studies began to probe the nature of vortex interactions in three dimensions. Verr´on et al. (1990) used a two layer quasi-geostrophic (QG) model to investigate the role of stratification in vortex merger. It was found that the merger distance initially increases√ from 3.2R for internal Rossby deformation radii ( g 0 H/f0 ) between 0 and 2, then relaxes back to this 2D limit for stronger stratification. In multi-layer models that better approximate continuous stratification, taller vortices exhibit stronger interactions and thus merge at greater distances (Von Hardenburg et al., 2000; Dritschel, 2002). These findings prove extremely useful for understanding vortex interactions in the real atmosphere. However, the highly idealized simulations used in many of these previous studies are restricted to quasi-geostrophic approximations, and may struggle to accurately represent atmospheric vortices such as tropical cyclones that have significant ageostrophic components. In this study, we investigate the interactions be-
2. Methodology The outline for this study is as follows: First, we define a TC-like control vortex and determine the critical merger distance. We then vary the TC depth and assess vortex interactions at a fixed time for two successive separation distances that are slightly larger than the determined merger distance. An observational approach is used to describe vortex interactions at these larger separation distances for progressively deeper vortices. Finally, we quantify vortex interactions using minimum sea level pressure for each experimental vortex. The model used for all simulations is version 1
3.4.1 of the Weather Research and Forecast Model with the Advanced Research dynamical core (WRFARW; Skamarock et al., 2008). WRF-ARW is a fully-compressible, non-hydrostatic model that is well-suited for atmospheric simulations. We use a singly-nested configuration: The outer domain is 6000 × 6000 km with 30km grid spacing and periodic lateral boundaries. A fixed nest with dimensions 990 × 990 km and 10km grid spacing is placed roughly in the center of the outer domain. All simulations are performed with 40 verticals, and on an f -plane at 20N. As we are only interested here in the dynamics of vortex interactions, we suppress any moist processes by turning off microphysics and cloud schemes. Furthermore, we do not employ radiation, surface or boundary layer schemes. Horizontal and vertical diffusion are constant, with diffusive constants of kh = 2000 m2 s−1 (horizontal) and kv = 5 m2 s−1 (vertical).
Figure 1: Tangential surface wind profile for a standard Rankine vortex (solid line) and the modified initial vortex (dashed line)
The vertical structure of the vortex wind field is controlled by two parameters, Lz and α, according to So as to facilitate comparisons with previous |z − zmax |α (3) V (r, z) = vs (r) × exp − studies using patch vortices, the initial vortices have αLαz a Rankine-like wind profile. The wind field of a stanThe effect of Lz and α is illustrated in Figure 2, dard Rankine vortex is which depicts various symmetric wind fields for an initial vortex. Lz represents the vortex depth in me ters, while α controls the rate at which the vortex r ≤ RM W vmax × RMr W wind field decays with height. Increasing α cuts off v0 (r) = (1) the vortex more sharply with height (Figures 2a and r > RM W vmax × RMr W 2b), and increasing Lz stretches the vortex deeper into the troposphere (Figures 2c and 2d).
where RM W is the radius of maximum winds (90 km), and vmax is the maximum wind speed (40 m/s). As a rough approximation of a tropical cyclone wind field, the vortex wind speed reaches a maximum at the surface. Following Nolan (2007), we then adjust the radial decay of the Rankine vortex so that its wind field does not extend to the periodic boundaries of the outer domain. The result is the modified vortex wind field vs (r),
vs (r) = v0 × exp
−
r Rcut
6 (2)
Figure 2: Symmetric wind field, V (r, z) for sample initial vortices with variable vertical structure: a) Lz = 4000m, α = 1.5; b) Lz = 4000m, α = 2.5; c) Lz = 5500m, α = 3.0, d) Lz = 6500m, α = 3.0 where Rcut is essentially the radius at which the
wind field is forced to zero (here we use Rcut = 800 km). The resulting radial wind profile at the surface The vortex for which the critical merger distance (z = 0) is depicted by the dashed line in Figure 1. is initially determined is referred to as the control 2
vortex. The vertical structure parameters of the control vortex are Lz = 4000m and α = 2.0, which, again, roughly approximates a typical tropical cyclone. For six progressively deeper experimental vortices, values of Lz and α must be carefully chosen so that the vortex wind field (1) is sufficiently confined to the troposphere, and (2) does not decay too rapidly with height, resulting in unstable overturning circulations in the upper vortex. This latter condition can be met by ensuring the largest negative symmetric PV is no more than 10% of the mean symmetric PV. A summary of the control and six experimental initial vortices satisfying these conditions appears in Table 1.
Figure 3: PV plots illustrating four vortex merger regimes: a) Diffusive, b) Non-core, c) Partial core, and d) Full core (1 PVU = 10−6 m2 s−1 K kg−1 )
Table 1: Vertical structure parameters for each initial vortex
Case Control Expt. 1 Expt. 2 Expt. 3 Expt. 4 Expt. 5 Expt. 6
Lz (km) 4.0 4.0 4.0 4.5 5.0 5.5 6.5
α 2.0 1.5 2.5 2.5 2.5 3.0 3.0
We justify the selection of 24 hours as the time for assessing vortex interactions based on the observation that vortex interactions after 24 hours are largely dominated by diffusive processes. With time, diffusion causes what we call diffusive merger to eventually look like non-core merger, and noncore merger to progress toward partial core merger. The diffusive constant kh = 2000 m2 s−1 used here is admittedly large, so the progression through these merger regimes may be unrealistically fast. Diffusion weakens the vortices as it draws them closer together and forces them to interact. Thus, we emphasize that we are most interested in full core merger because it is the only class of interactions that is not achieved by diffusion with enough time.
The pressure and temperature fields are iteratively adjusted so that the vortex is in gradient-wind and hydrostatic balance (Nolan et al., 2001). We then add two “copies” of the balanced vortices along a line of constant latitude to a domain initialized with an atmospheric sounding typical of the tropical Atlantic basin during hurricane season (Dunion and Marron, 2008). 3. Results In order to determine a critical merger distance, In determining the critical merger distance, we claswe first identify four classes of interactions based sify vortex interactions at three different pressure on plots of PV on constant pressure surfaces. These levels (900, 850, and 700 hPa) after 24 hours for four classes are (1) Diffusive merger, in which the separation distances between 240 and 520 km. The outer regions of the vortex are connected but the ini- critical distance for each class of interactions is the tial vortex cores remain intact; (2) Non-core merger, separation distance at which a particular regime is in which the outer vortex regions are fully-merged, first observed at either of the three pressure levels. but two cores are still distinguishable; (3) Partial core merger, in which there is a fully-merged outer a. Control Vortex region, significant straining out, and no evidence of Using the above criteria, the critical distances (dc ) the two initial cores; and (4) Full core or complete for each regime for the control vortex are summamerger, resulting in a monopole vortex with mini- rized in Table 2. The critical distance for full core merger is very mal straining out. Examples of the four regimes are depicted in Figure 3 using PV on constant pressure close to the 2D limit. Here, the radius of our initial vortex is the RM W = 90 km, so 3.2R = 288 surfaces. 3
km. Note that as a result of initializing vortices on von Hardenburg et al., 2000; Dritschel, 2002). the 30-km grid, it is possible that the true critical distance is actually between 270 km and 300 km. Table 2: Critical separation distances for each type of merger of two control vortices
Regime Full Core Partial Core Non-Core Diffusive
dc (km) 270 300 360 480
b. Deeper Vortices Having established a benchmark critical merger distance for the control vortex, we now examine how this distance changes for progressively deeper vortices. Simulations for the experimental vortices are carried out at separation distances of 300km and 360km, just beyond the critical merger distance for the control vortex. We will first present the results for three progressively deeper vortices (expts. 2, 4 and 6) at 300km separation. Plane views and vertical cross sections of PV after 24 hours are shown in Figure 4. The left column of Figure 4 depicts PV along constant pressure surfaces (only the nests are depicted). The deepest initial vortices (expts. 4 & 6, bottom two rows) can indeed achieve full-core merger within 24 hours at a separation distance beyond the 2D limit (300 km = 3.33R). For the control and the shallower expt. 2 vortex, we observe only partial core merger by 24 hours. The corresponding vertical cross-sections (right column of Figure 4) reveal some interesting features for progressively deeper vortices. First, even for what is classified as partial core merger using the plane-views of the shallowest vortices, we still see signatures of two cores confined near the surface. Moreover, merger tends to take place higher in the atmosphere for deeper vortices, with distinct separation between the initial cores evident at the surface and upper levels (this is especially so for the deepest expt. 6 vortices). As will be discussed later, this behavior may be an artifact of non-uniform stratification in our domain. Additional features evident in the vertical PV cross-sections include enveloping filaments for the deepest vortex, as well as vortex flattening associated with merger (not shown). Both of these phenomena were also observed in previous studies (e.g.
Figure 4: PV plane views on the indicated pressure levels (left column), and vertical cross-sections (right column) at t=24 hours. The top row is the control, the second row is expt. 2 (Lz = 4000, α = 2.5), the third row is expt. 4 (Lz = 5000, α = 2.5), and the bottom row is the deepest vortex, expt. 6 (Lz = 6500, α = 3.0)
Figure 5 summarizes the merger regimes observed at 24 hours for both the 300km and 360km separation distances. The tendency for merger to occur at higher levels for deeper vortices is evident in both plots, as the strongest interactions occur at 900-hPa (blue line) for shallow vortices, and at 700-hPa (red line) for deeper vortices. The right panel indicates that at 360km separation, full core merger is not achieved for even the deepest vortex (corresponding PV plots for 360km separation are not shown here). However, it is certainly possible that yet deeper vortices than those tested here could achieve full core merger at 360km separation. 4
Figure 5: Summary of merger regimes after 24 hours for control + experimental vortices at 300 km separation (left) and 360 km separation (right). Merger regimes are diagnosed at three different pressure levels: 900 hPa (blue line), 850 hPa (green line), and 700 hPa (red line)
c. Sea Level Pressure Tendency The minimum sea level pressure (MSLP) in the domain at each time step is used to quantify vortex interactions for each experiment. Figure 6 illustrates that the MSLP for all vortices follows roughly the same trajectory initially. That is, there are steady pressure rises through about 4 hours for 300-km separation, and through about 10 hours for 360km separation. Thereafter, the MSLP tendency exhibits a strong dependency on the depth of the vortex. Deeper vortices experience marked pressure falls, while the pressure for shallower vortices either levels off, or continues to rise at a slower rate.
4. Discussion and Conclusions This study attempts to improve our understanding of vortex interactions in the real atmosphere using idealized simulations from a fully-compressible atmospheric model (WRF-ARW). Our first significant finding is that the critical distance for full core merger of the control TC-like vortex is roughly the 2D limit of 3.2R. Furthermore, we have shown that deeper vortices can achieve full core merger at distances beyond this limit. This result agrees with Dritschel (2002), who found that the 2D critical merger distance is not actually an upper limit for particularly tall 3D vortices.
Figure 6: Minimum sea level pressure vs. time for 300km separation (top) and 360km separation (bottom). The black line represents the control vortex, and colored lines represent progressively deeper vortices (yellow = shallowest, blue = deepest)
We can understand the effect this has on our PV distribution using the definition of PV for a comA unique finding in this work is the tendency pressible fluid in the absence of heat sources, for merger to occur at higher levels for deeper vorω~a · ∇θ tices. This could potentially be explained by the q= (4) ρ fact that the environment has non-uniform stratification. Specifically, the tropical atmospheric sound- where ω~a is the absolute vorticity, θ is potential teming used to initialize the domain is such that the perature and ρ is density. Varying N 2 in the vertical Brunt-V¨ ais¨ al¨ a frequency (N 2 ) varies with height. changes the PV of the prescribed vortex through the 5
∇θ term. Therefore, isolating the PV field to that only associated with the vortex would require prescribing constant background stratification. This is the formulation used in previous studies of vortex merger in QG fluids. One particularly interesting finding is a resistance to full core merger throughout the depth of the vortex. This is evident in vertical PV cross sections for the deepest vortex (lower right panel of Figure 4), where two cores are still evident at 24 hours even after strong interactions. One possible explanation for this has to do with the nature of 3D filamentation and axisymmetrization. Filaments ejected from the upper and lower parts of a 3D vortex eventually cause the vortex to become flatter with time. Vortex flattening is more dramatic for deeper vortices, which we observed to experience greater filamentation (Verr´ on et al. (1990) also confirm greater filamentation for deeper vortices in a two layer model). A shallower vortex will have more local interactions, so vortex flattening may be associated with weakening interactions between the vortices that ultimately acts to preserve their initial inner core structures. Also noteworthy is the strong dependence of minimum sea level pressure tendency on vortex depth. An investigation of the PV snapshots both prior to and after the time when MSLP tendencies diverge (Figure 6) suggests that pressure falls are associated with strong vorticity mixing between the vortex cores. A relationship between vorticity mixing and pressure was established by Kossin and Schubert (2001) (hereafter KS01). In a study using a 2D barotropic model, they found that the merger of mesovortices resulting from the unstable breakdown of an annular vorticity distribution can cause dramatic pressure falls as large as 50 hPa. This can be physically explained by assuming the vortex wind field is approximately in cyclostrophic balance: v(r)2 1 ∂P =− r ρ ∂r
corresponding pressure falls. That said, the net pressure falls we observe are only 5 hPa for the deepest vortex, which is an order of magnitude smaller than those observed by KS01. This difference could be due their use of a 2D model, which inherently has stronger interactions between vortices, and thus greater vorticity mixing. Consequently, the pressure falls associated with merging mesovortices during tropical cyclogenesis in the real atmosphere may actually be less significant than originally predicted.
Acknowledgements The author would like to thank Dr. David S. Nolan for numerous insightful discussions and interpretations of these results. Also, thanks to Dr. Nolan and Matt Onderlinde for their assistance configuring WRF-ARW for prescribed vortex initialization. All computations were performed using the University of Miami’s High Performance Computing clusters.
References Dritschel, D.G., 2002. Vortex merger in rotating stratified flows. J. Fluid Mech., 455, 83-101. Dunion, J.P., and Marron, C.S, 2007. A Reexamination of the Jordan Mean Tropical Sounding Based on Awareness of the Saharan Air Layer: Results from 2002. J. Climate, 21, 5242-5253. Kossin, J.P. and Schubert, W.S. 2001. Mesovortices, Polygonal Flow Patterns, and Rapid Pressure Falls in Hurricane-Like Vortices. J. Atmos. Sci., 58, 2196-2209. Melander, M.V., McWilliams, J.C., and Zabusky, N.J., 1987. Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech., 178, 137-159. Melander, M.V., Zabusky, N.J., and McWilliams, J.C., 1988. Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech., 195, 303-340. Nolan, D.S. 2007. What is the trigger for tropical cyclogensis? Aust. Met. Mag., 56, 241-266. Nolan, D.S., Montgomery, M.T., and Grasso, L.D. 2001. The wavenumber one instability and trochoidal motion of hurricane-like vortices. J. Atmos. Sci., 58, 3243-3270. Skamarock, W.C., Klemp, J.B., Dudhia, J., Gill, D.O., Barker, D.M., Duda, M.G., Huang, X-Y., Wang, W., and Powers, J.G. 2008. A Description of the Advanced Research WRF Version 3. NCAR Technical Note 475+STR, 113 pp. Verr´ on, J., Hopfinger, E.J, and McWilliams, J.C. 1990. Sensitivity to initial conditions in the merging of twolayer baroclinic vortices. Phys. Fluids, A 2, 886-889. Von Hardenburg, J., McWilliams, J.C., Provenzale, A., Shchepetkin, A., and Weiss, J.B. 2000. Vortex merging in quasi-geostrophic flows. J. Fluid Mech., 412, 331-353.
(5)
The tangential wind profile v(r) in the inner part of an axisymmetrizing vortex steepens as the vortex core better approximates solid body rotation. This acceleration of the tangential wind is supported by an increasing radial pressure gradient, which can only occur if pressure decreases in the core of the vortex. Considering vortex merger is an axisymmetrization process, we therefore expect 6
1. Introduction
tween two symmetric tropical-cyclone like vortices without the constraints of quasi-geostrophy. In order to simulate vortex interactions in the real atmosphere, we enlist the fully-compressible Weather Research and Forecast (WRF) model. A variety of experimental vortices of variable depth are simulated in order to understand the role of vortex depth in the merger process. Following von Hardenburg et al. (2000), we can anticipate how the critical merger distance might behave in 2D vs. 3D stratified flow using the Green’s Function. In 2D, the induced flow by a point vortex decays radially as log(r). However, in 3D stratified flow, the induced flow by a point of potential vorticity (PV) decays as a modified Bessel function of the second kind, implying more localized interactions. 3D vortex interactions are also sensitive to the degree of stratification: Stronger stratification increases the deformation radius and thus the extent of the induced flow field by a vortex. As taller vortices better approximate a 2D vortex (which has infinite depth), we expect the critical merger distance to increase with vortex depth. Both von Hardenburg et al. (2000) and Dritschel (2002) encounter such behavior for taller vortices, but some confusion arises on whether the 2D barotropic limit is indeed a limit on critical merger distance. It is precisely this sort of discrepancy that we wish to clarify in this study.
Vortex merger and interaction has been studied extensively from a theoretical fluid dynamics perspective. How such interactions unfold in the real atmosphere, however, remains unclear. Nonetheless, twodimensional vortex dynamics is a natural starting point from which to understand atmospheric vortex interactions. Melander et al. (1987) first illustrated the so-called axisymmetrization of an elliptical twodimensional patch vortex as a process in which filamentation of the vortex core disrupts the symmetry of the flow field. As a result, stream lines can cross vorticity contours so as to decrease the aspect ratio of the elliptical vortex, ultimately forcing it toward near axisymmetry. Melander et al. (1988) considered vortex merger an axisymmetrization process, and established a critical merger distance for two patch vortices of the same size and strength (i.e. “symmetric” vortices) ≈ 3.2 times the initial vortex radius (R). We will henceforth refer to 3.2R as the 2D limit for vortex merger. Subsequent studies began to probe the nature of vortex interactions in three dimensions. Verr´on et al. (1990) used a two layer quasi-geostrophic (QG) model to investigate the role of stratification in vortex merger. It was found that the merger distance initially increases√ from 3.2R for internal Rossby deformation radii ( g 0 H/f0 ) between 0 and 2, then relaxes back to this 2D limit for stronger stratification. In multi-layer models that better approximate continuous stratification, taller vortices exhibit stronger interactions and thus merge at greater distances (Von Hardenburg et al., 2000; Dritschel, 2002). These findings prove extremely useful for understanding vortex interactions in the real atmosphere. However, the highly idealized simulations used in many of these previous studies are restricted to quasi-geostrophic approximations, and may struggle to accurately represent atmospheric vortices such as tropical cyclones that have significant ageostrophic components. In this study, we investigate the interactions be-
2. Methodology The outline for this study is as follows: First, we define a TC-like control vortex and determine the critical merger distance. We then vary the TC depth and assess vortex interactions at a fixed time for two successive separation distances that are slightly larger than the determined merger distance. An observational approach is used to describe vortex interactions at these larger separation distances for progressively deeper vortices. Finally, we quantify vortex interactions using minimum sea level pressure for each experimental vortex. The model used for all simulations is version 1
3.4.1 of the Weather Research and Forecast Model with the Advanced Research dynamical core (WRFARW; Skamarock et al., 2008). WRF-ARW is a fully-compressible, non-hydrostatic model that is well-suited for atmospheric simulations. We use a singly-nested configuration: The outer domain is 6000 × 6000 km with 30km grid spacing and periodic lateral boundaries. A fixed nest with dimensions 990 × 990 km and 10km grid spacing is placed roughly in the center of the outer domain. All simulations are performed with 40 verticals, and on an f -plane at 20N. As we are only interested here in the dynamics of vortex interactions, we suppress any moist processes by turning off microphysics and cloud schemes. Furthermore, we do not employ radiation, surface or boundary layer schemes. Horizontal and vertical diffusion are constant, with diffusive constants of kh = 2000 m2 s−1 (horizontal) and kv = 5 m2 s−1 (vertical).
Figure 1: Tangential surface wind profile for a standard Rankine vortex (solid line) and the modified initial vortex (dashed line)
The vertical structure of the vortex wind field is controlled by two parameters, Lz and α, according to So as to facilitate comparisons with previous |z − zmax |α (3) V (r, z) = vs (r) × exp − studies using patch vortices, the initial vortices have αLαz a Rankine-like wind profile. The wind field of a stanThe effect of Lz and α is illustrated in Figure 2, dard Rankine vortex is which depicts various symmetric wind fields for an initial vortex. Lz represents the vortex depth in me ters, while α controls the rate at which the vortex r ≤ RM W vmax × RMr W wind field decays with height. Increasing α cuts off v0 (r) = (1) the vortex more sharply with height (Figures 2a and r > RM W vmax × RMr W 2b), and increasing Lz stretches the vortex deeper into the troposphere (Figures 2c and 2d).
where RM W is the radius of maximum winds (90 km), and vmax is the maximum wind speed (40 m/s). As a rough approximation of a tropical cyclone wind field, the vortex wind speed reaches a maximum at the surface. Following Nolan (2007), we then adjust the radial decay of the Rankine vortex so that its wind field does not extend to the periodic boundaries of the outer domain. The result is the modified vortex wind field vs (r),
vs (r) = v0 × exp
−
r Rcut
6 (2)
Figure 2: Symmetric wind field, V (r, z) for sample initial vortices with variable vertical structure: a) Lz = 4000m, α = 1.5; b) Lz = 4000m, α = 2.5; c) Lz = 5500m, α = 3.0, d) Lz = 6500m, α = 3.0 where Rcut is essentially the radius at which the
wind field is forced to zero (here we use Rcut = 800 km). The resulting radial wind profile at the surface The vortex for which the critical merger distance (z = 0) is depicted by the dashed line in Figure 1. is initially determined is referred to as the control 2
vortex. The vertical structure parameters of the control vortex are Lz = 4000m and α = 2.0, which, again, roughly approximates a typical tropical cyclone. For six progressively deeper experimental vortices, values of Lz and α must be carefully chosen so that the vortex wind field (1) is sufficiently confined to the troposphere, and (2) does not decay too rapidly with height, resulting in unstable overturning circulations in the upper vortex. This latter condition can be met by ensuring the largest negative symmetric PV is no more than 10% of the mean symmetric PV. A summary of the control and six experimental initial vortices satisfying these conditions appears in Table 1.
Figure 3: PV plots illustrating four vortex merger regimes: a) Diffusive, b) Non-core, c) Partial core, and d) Full core (1 PVU = 10−6 m2 s−1 K kg−1 )
Table 1: Vertical structure parameters for each initial vortex
Case Control Expt. 1 Expt. 2 Expt. 3 Expt. 4 Expt. 5 Expt. 6
Lz (km) 4.0 4.0 4.0 4.5 5.0 5.5 6.5
α 2.0 1.5 2.5 2.5 2.5 3.0 3.0
We justify the selection of 24 hours as the time for assessing vortex interactions based on the observation that vortex interactions after 24 hours are largely dominated by diffusive processes. With time, diffusion causes what we call diffusive merger to eventually look like non-core merger, and noncore merger to progress toward partial core merger. The diffusive constant kh = 2000 m2 s−1 used here is admittedly large, so the progression through these merger regimes may be unrealistically fast. Diffusion weakens the vortices as it draws them closer together and forces them to interact. Thus, we emphasize that we are most interested in full core merger because it is the only class of interactions that is not achieved by diffusion with enough time.
The pressure and temperature fields are iteratively adjusted so that the vortex is in gradient-wind and hydrostatic balance (Nolan et al., 2001). We then add two “copies” of the balanced vortices along a line of constant latitude to a domain initialized with an atmospheric sounding typical of the tropical Atlantic basin during hurricane season (Dunion and Marron, 2008). 3. Results In order to determine a critical merger distance, In determining the critical merger distance, we claswe first identify four classes of interactions based sify vortex interactions at three different pressure on plots of PV on constant pressure surfaces. These levels (900, 850, and 700 hPa) after 24 hours for four classes are (1) Diffusive merger, in which the separation distances between 240 and 520 km. The outer regions of the vortex are connected but the ini- critical distance for each class of interactions is the tial vortex cores remain intact; (2) Non-core merger, separation distance at which a particular regime is in which the outer vortex regions are fully-merged, first observed at either of the three pressure levels. but two cores are still distinguishable; (3) Partial core merger, in which there is a fully-merged outer a. Control Vortex region, significant straining out, and no evidence of Using the above criteria, the critical distances (dc ) the two initial cores; and (4) Full core or complete for each regime for the control vortex are summamerger, resulting in a monopole vortex with mini- rized in Table 2. The critical distance for full core merger is very mal straining out. Examples of the four regimes are depicted in Figure 3 using PV on constant pressure close to the 2D limit. Here, the radius of our initial vortex is the RM W = 90 km, so 3.2R = 288 surfaces. 3
km. Note that as a result of initializing vortices on von Hardenburg et al., 2000; Dritschel, 2002). the 30-km grid, it is possible that the true critical distance is actually between 270 km and 300 km. Table 2: Critical separation distances for each type of merger of two control vortices
Regime Full Core Partial Core Non-Core Diffusive
dc (km) 270 300 360 480
b. Deeper Vortices Having established a benchmark critical merger distance for the control vortex, we now examine how this distance changes for progressively deeper vortices. Simulations for the experimental vortices are carried out at separation distances of 300km and 360km, just beyond the critical merger distance for the control vortex. We will first present the results for three progressively deeper vortices (expts. 2, 4 and 6) at 300km separation. Plane views and vertical cross sections of PV after 24 hours are shown in Figure 4. The left column of Figure 4 depicts PV along constant pressure surfaces (only the nests are depicted). The deepest initial vortices (expts. 4 & 6, bottom two rows) can indeed achieve full-core merger within 24 hours at a separation distance beyond the 2D limit (300 km = 3.33R). For the control and the shallower expt. 2 vortex, we observe only partial core merger by 24 hours. The corresponding vertical cross-sections (right column of Figure 4) reveal some interesting features for progressively deeper vortices. First, even for what is classified as partial core merger using the plane-views of the shallowest vortices, we still see signatures of two cores confined near the surface. Moreover, merger tends to take place higher in the atmosphere for deeper vortices, with distinct separation between the initial cores evident at the surface and upper levels (this is especially so for the deepest expt. 6 vortices). As will be discussed later, this behavior may be an artifact of non-uniform stratification in our domain. Additional features evident in the vertical PV cross-sections include enveloping filaments for the deepest vortex, as well as vortex flattening associated with merger (not shown). Both of these phenomena were also observed in previous studies (e.g.
Figure 4: PV plane views on the indicated pressure levels (left column), and vertical cross-sections (right column) at t=24 hours. The top row is the control, the second row is expt. 2 (Lz = 4000, α = 2.5), the third row is expt. 4 (Lz = 5000, α = 2.5), and the bottom row is the deepest vortex, expt. 6 (Lz = 6500, α = 3.0)
Figure 5 summarizes the merger regimes observed at 24 hours for both the 300km and 360km separation distances. The tendency for merger to occur at higher levels for deeper vortices is evident in both plots, as the strongest interactions occur at 900-hPa (blue line) for shallow vortices, and at 700-hPa (red line) for deeper vortices. The right panel indicates that at 360km separation, full core merger is not achieved for even the deepest vortex (corresponding PV plots for 360km separation are not shown here). However, it is certainly possible that yet deeper vortices than those tested here could achieve full core merger at 360km separation. 4
Figure 5: Summary of merger regimes after 24 hours for control + experimental vortices at 300 km separation (left) and 360 km separation (right). Merger regimes are diagnosed at three different pressure levels: 900 hPa (blue line), 850 hPa (green line), and 700 hPa (red line)
c. Sea Level Pressure Tendency The minimum sea level pressure (MSLP) in the domain at each time step is used to quantify vortex interactions for each experiment. Figure 6 illustrates that the MSLP for all vortices follows roughly the same trajectory initially. That is, there are steady pressure rises through about 4 hours for 300-km separation, and through about 10 hours for 360km separation. Thereafter, the MSLP tendency exhibits a strong dependency on the depth of the vortex. Deeper vortices experience marked pressure falls, while the pressure for shallower vortices either levels off, or continues to rise at a slower rate.
4. Discussion and Conclusions This study attempts to improve our understanding of vortex interactions in the real atmosphere using idealized simulations from a fully-compressible atmospheric model (WRF-ARW). Our first significant finding is that the critical distance for full core merger of the control TC-like vortex is roughly the 2D limit of 3.2R. Furthermore, we have shown that deeper vortices can achieve full core merger at distances beyond this limit. This result agrees with Dritschel (2002), who found that the 2D critical merger distance is not actually an upper limit for particularly tall 3D vortices.
Figure 6: Minimum sea level pressure vs. time for 300km separation (top) and 360km separation (bottom). The black line represents the control vortex, and colored lines represent progressively deeper vortices (yellow = shallowest, blue = deepest)
We can understand the effect this has on our PV distribution using the definition of PV for a comA unique finding in this work is the tendency pressible fluid in the absence of heat sources, for merger to occur at higher levels for deeper vorω~a · ∇θ tices. This could potentially be explained by the q= (4) ρ fact that the environment has non-uniform stratification. Specifically, the tropical atmospheric sound- where ω~a is the absolute vorticity, θ is potential teming used to initialize the domain is such that the perature and ρ is density. Varying N 2 in the vertical Brunt-V¨ ais¨ al¨ a frequency (N 2 ) varies with height. changes the PV of the prescribed vortex through the 5
∇θ term. Therefore, isolating the PV field to that only associated with the vortex would require prescribing constant background stratification. This is the formulation used in previous studies of vortex merger in QG fluids. One particularly interesting finding is a resistance to full core merger throughout the depth of the vortex. This is evident in vertical PV cross sections for the deepest vortex (lower right panel of Figure 4), where two cores are still evident at 24 hours even after strong interactions. One possible explanation for this has to do with the nature of 3D filamentation and axisymmetrization. Filaments ejected from the upper and lower parts of a 3D vortex eventually cause the vortex to become flatter with time. Vortex flattening is more dramatic for deeper vortices, which we observed to experience greater filamentation (Verr´ on et al. (1990) also confirm greater filamentation for deeper vortices in a two layer model). A shallower vortex will have more local interactions, so vortex flattening may be associated with weakening interactions between the vortices that ultimately acts to preserve their initial inner core structures. Also noteworthy is the strong dependence of minimum sea level pressure tendency on vortex depth. An investigation of the PV snapshots both prior to and after the time when MSLP tendencies diverge (Figure 6) suggests that pressure falls are associated with strong vorticity mixing between the vortex cores. A relationship between vorticity mixing and pressure was established by Kossin and Schubert (2001) (hereafter KS01). In a study using a 2D barotropic model, they found that the merger of mesovortices resulting from the unstable breakdown of an annular vorticity distribution can cause dramatic pressure falls as large as 50 hPa. This can be physically explained by assuming the vortex wind field is approximately in cyclostrophic balance: v(r)2 1 ∂P =− r ρ ∂r
corresponding pressure falls. That said, the net pressure falls we observe are only 5 hPa for the deepest vortex, which is an order of magnitude smaller than those observed by KS01. This difference could be due their use of a 2D model, which inherently has stronger interactions between vortices, and thus greater vorticity mixing. Consequently, the pressure falls associated with merging mesovortices during tropical cyclogenesis in the real atmosphere may actually be less significant than originally predicted.
Acknowledgements The author would like to thank Dr. David S. Nolan for numerous insightful discussions and interpretations of these results. Also, thanks to Dr. Nolan and Matt Onderlinde for their assistance configuring WRF-ARW for prescribed vortex initialization. All computations were performed using the University of Miami’s High Performance Computing clusters.
References Dritschel, D.G., 2002. Vortex merger in rotating stratified flows. J. Fluid Mech., 455, 83-101. Dunion, J.P., and Marron, C.S, 2007. A Reexamination of the Jordan Mean Tropical Sounding Based on Awareness of the Saharan Air Layer: Results from 2002. J. Climate, 21, 5242-5253. Kossin, J.P. and Schubert, W.S. 2001. Mesovortices, Polygonal Flow Patterns, and Rapid Pressure Falls in Hurricane-Like Vortices. J. Atmos. Sci., 58, 2196-2209. Melander, M.V., McWilliams, J.C., and Zabusky, N.J., 1987. Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech., 178, 137-159. Melander, M.V., Zabusky, N.J., and McWilliams, J.C., 1988. Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech., 195, 303-340. Nolan, D.S. 2007. What is the trigger for tropical cyclogensis? Aust. Met. Mag., 56, 241-266. Nolan, D.S., Montgomery, M.T., and Grasso, L.D. 2001. The wavenumber one instability and trochoidal motion of hurricane-like vortices. J. Atmos. Sci., 58, 3243-3270. Skamarock, W.C., Klemp, J.B., Dudhia, J., Gill, D.O., Barker, D.M., Duda, M.G., Huang, X-Y., Wang, W., and Powers, J.G. 2008. A Description of the Advanced Research WRF Version 3. NCAR Technical Note 475+STR, 113 pp. Verr´ on, J., Hopfinger, E.J, and McWilliams, J.C. 1990. Sensitivity to initial conditions in the merging of twolayer baroclinic vortices. Phys. Fluids, A 2, 886-889. Von Hardenburg, J., McWilliams, J.C., Provenzale, A., Shchepetkin, A., and Weiss, J.B. 2000. Vortex merging in quasi-geostrophic flows. J. Fluid Mech., 412, 331-353.
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The tangential wind profile v(r) in the inner part of an axisymmetrizing vortex steepens as the vortex core better approximates solid body rotation. This acceleration of the tangential wind is supported by an increasing radial pressure gradient, which can only occur if pressure decreases in the core of the vortex. Considering vortex merger is an axisymmetrization process, we therefore expect 6
